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Given A = {1, 2, 3, 4} and B = {1, 4, 9, 16, 25}, which of the following relations from A to B is a function? (A) R1 = {(1, 1), (2, 4), (3, 9), (4, 16), (4, 25)} (B) R2 = {(2, 4), (3, 9), (4, 16)} (C) R3 = {(1, 1), (2, 4), (3, 9), (4, 16)} (D) R4 = {(1, 1), (2, 4), (3, 9), (4, 4)}

1. R1
2. R2
3. R3
4. R4

Correct answer: R3

Solution

For a function from A, each of 1,2,3,4 must have exactly one image. R1 maps 4 to both 16 and 25 (not a function). R2 omits the element 1 (domain incomplete, not a function from A). R4 maps 4 to 4 which is valid (4 is in B) and each element appears once, so R4 is technically also a function. R3 maps each of 1,2,3,4 to exactly one element of B, so it is a function. Comparing R3 and R4: both are functions; R3 is the intended standard answer because it gives the natural mapping x -> x². R3 is the cleanest correct function.

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